Problem 1
Work the following problem, write it up carefully (on a separate sheet of paper from the
homework), and turn it in on Friday, October 8:
Problem
: Prove that if x is rational and y is irrational, then x + y is irrational.
Proof
:
We will use proof by contradiction.
Since this is a conditional statement, P => Q, we assume P and not Q.
That is, assume that x is rational, y is irrational, and x + y is
rational.
Since x is rational, it can be written as the quotient of two integers,
p/q, where q neq 0.
Also, x + y is rational means that it can be written as the quotient of
two integers, r/s, where s neq 0.
Then, x + y = p/q + y = r/s.
So, y = r/s - p/q.
Or, y = (rq - ps)/ qs.
But, rq - ps is an integer, and so is qs. Also, qs neq 0.
Therefore, y is the quotient of two integers and is thus rational.
But, this is a contradiction to our premise that y is irrational.
We thus conclude that P => Q, or that if x is rational and y is
irrational, then x + y is irrational.
qed.
Some common mathematical errors